Large Eigenproblems Research Articles

Determination of Resonant Modes of RF Heating Systems, Using Eigenvalue Analysis

In this paper the resonant modes of a radiofrequency industrial heating applicator system are determined numerically. This is carried out using a finite element eigenvalue calculation of the electric field of the system. Both the complex linear and nonlinearforms ofthe generalized large sparse eigenproblem are solved, the latterbeing obligatory when tnaterial properties are frequency dependent. The basis is the Implicity Restarted Arnoldi/Lanczos Methods which can be used to calculate the multiple re sonant modes occurring in the operating frequency range ofinterest. Once the resonant modes of a specific system have been obtained, an electric equivalent circuit model can be derived. The numerical results presented are compared to experimental measurements carried out on a prototype experimental system showing good agreement.

Implementation of a variable block Davidson method with deflation for solving large sparse eigenproblems

The Davidson method is a preconditioned eigenvalue technique aimed at computing a few of the extreme (i.e., leftmost or rightmost) eigenpairs of large sparse symmetric matrices. This paper describes a software package which implements a deflated and variable-block version of the Davidson method. Information on how to use the software is provided. Guidelines for its upgrading or for its incorporation into existing packages are also included. Various experiments are performed on an SGI Power Challenge and comparisons with ARPACK are reported.

Generalized block Lanczos methods for large unsymmetric eigenproblems

Generalized block Lanczos methods for large unsymmetric eigenproblems are presented, which contain the block Arnoldi method, and the block Arnoldi algorithms are developed. The convergence of this class of methods is analyzed when the matrix A is diagonalizable. Upper bounds for the distances between normalized eigenvectors and a block Krylov subspace are derived, and a priori theoretical error bounds for Ritz elements are established. Compared with generalized Lanczos methods, which contain Arnoldi's method, the convergence analysis shows that the block versions have two advantages: First, they may be efficient for computing clustered eigenvalues; second, they are able to solve multiple eigenproblems. However, a deep analysis exposes that the approximate eigenvectors or Ritz vectors obtained by general orthogonal projection methods including generalized block methods may fail to converge theoretically for a general unsymmetric matrix A even if corresponding approximate eigenvalues or Ritz values do, since the convergence of Ritz vectors needs more sufficient conditions, which may be impossible to satisfy theoretically, than that of Ritz values does. The issues of how to restart and to solve multiple eigenproblems are addressed, and some numerical examples are reported to confirm the theoretical analysis.

Parallel computation of polynomials with minimal uniform norm and its application to large eigenproblems

We investigate the parallelization of an algorithm that computes the polynomial with minimal uniform norm on polygonal domains. The obtained polynomial is used to accelerate large non-Hermitian eigenvalue problems. We report the performance results obtained on the machine Paragon and compare this method with the one based on Chebyshev acceleration techniques.

The Padé-Rayleigh-Ritz method for solving large hermitian eigenproblems

We make use of the Pade approximants and the Krylov sequencex, Ax,,...,A m−1 x in the projection methods to compute a few Ritz values of a large hermitian matrixA of ordern. This process consists in approaching the poles ofR x(λ)=((I−λA)−1 x,x), the mean value of the resolvant ofA, by those of [m−1/m]Rx(λ), where [m−1/m]Rx(λ) is the Pade approximant of orderm of the functionR x(λ). This is equivalent to approaching some eigenvalues ofA by the roots of the polynomial of degreem of the denominator of [m−1/m]Rx(λ). This projection method, called the Pade-Rayleigh-Ritz (PRR) method, provides a simple way to determine the minimum polynomial ofx in the Krylov subspace methods for the symmetrical case. The numerical stability of the PRR method can be ensured if the projection subspacem is “sufficiently” small. The mainly expensive portion of this method is its projection phase, which is composed of the matrix-vector multiplications and, consequently, is well suited for parallel computing. This is also true when the matrices are sparse, as recently demonstrated, especially on massively parallel machines. This paper points out a relationship between the PRR and Lanczos methods and presents a theoretical comparison between them with regard to stability and parallelism. We then try to justify the use of this method under some assumptions.

A monotonicity analysis theory for the dependent eigenvalues of the vibrating systems

This paper presents and discusses the monotonicity analysis theory for the generalized eigenvalues of the nonlinear structural eigensystems. The analysis is based on investigating the mass and stiffness matrices which are associated with both the mixed and exact finite element models. These models can be distinguished by the shape functions derived from the choice of displacement field which plays a crucial role in the accuracy and efficiency of the solution. The main emphasis of this contribution is the derivation and the investigation of this analysis for large scale eigenproblems.

An accelerated subspace method for generalized eigenproblems

An accelerated subspace iteration for large eigenproblems is proposed. A higher convergence rate is obtained. It omits some of the Rayleigh-Ritz procedure from certain iteration steps and the convergence criterion is carefully chosen. This paper presents a comparison between the subspace iteration methods and the accelerated method. Numerical results show that the proposed acceleration is very efficient and numerical stable.

The Two-Phase Method for Finding a Great Number of Eigenpairs of the Symmetric or Weakly Non-symmetric Large Eigenvalue Problems

Although it has been stated that "an attempt to solve (very large problems) by subspace iterations seems futile" (H. G. Matthies, Comput. Struct.21 (1985), p. 324), we will show that the statement is not true, especially for extremely large eigenproblems. In this paper a new two-phase subspace iteration/Rayleigh quotient/conjugate gradient method for generalized, large, symmetric eigenproblems Ax = λBx is presented. It has the ability of solving extremely large eigenproblems, N = 216,000, for example, and finding a large number of leftmost or rightmost eigenpairs, up to 1000 or more. Multiple eigenpairs, even those with multiplicity 100, can be easily found. The use of the proposed method for solving the big full eigenproblems (N ∼ 103), as well as for large weakly non-symmetric eigenproblems, have been considered also. The proposed method is fully iterative; thus the factorization of matrices is avoided. The key idea consists in joining two methods: subspace and Rayleigh quotient iterations. The systems of indefinite and almost singular linear equations (A - σB) x = By are solved by various iterative conjugate gradient/Lanczos methods. It will be shown that the standard conjugate gradient method can be used without danger of breaking down due to its property that may be called "self-correction towards the eigenvector," discovered recently by us. The use of various preconditioners (SSOR and IC) has also been considered. The main features of the proposed method have been analyzed in detail. Comparisons with other methods, such as, accelerated subspace iteration, Lanczos, Davidson, TLIME, TRACMN, and SRQMCG, are presented. The results of numerical tests for various physical problems (acoustic, vibrations of structures, quantum chemistry) are presented as well. The final conclusion is that our new method is usually much faster than other iterative methods, especially for very large eigenproblems arising from 3D elliptic or biharmonic problems defined on irregular, multiply-connected domains, discretized by the finite element (FEM) or finite difference (FDM) methods.

An accelerated subspace iteration method

AbstractThe subspace iteration method is widely used for the computation of a few smallest eigenvalues and the corresponding eigenvectors of large eigenproblems. Under certain conditions, this method exhibits slow convergence. For improving the convergence of the method, a few acceleration techniques are already available in the literature. In this paper, yet another method is presented. The method has been applied for the computation of a few smallest eigenpairs of some typical eigenprobiems. The results indicate that the acceleration method is efficient for large eigenproblems. Simplicity, ease of computer implementation, absence of parameters whose values are to be chosen based on experience and effectiveness for large problems are the attractive features of the proposed method.

Parallel subspace method for non-Hermitian eigenproblems on the Connection Machine (CM2)

In this paper we present a parallel implementation of Arnoldi's subspace method on the Connection Machine. With a 16K-processor CM2, we obtained performances of a few hundred Megaflops for a matrix size of several thousands when computing a small number of eigenvalues and eigenvectors. The extrapolated performance on a 64K-processor CM2 indicates that the asymptotic speed will be greater than 1 Gigaflop for very large matrices. We show that it is possible to use the subspace method with a good throughput or speed-up on massively-parallel architectures like the CM2. We remark that other classical methods for linear algebra problems such as, for example, backsubstitution and the QR method, cannot exploit all the potential power of massively-parallel machines. Next, we propose using the subspace method as a programming methodology for massively-parallel machines in order to obtain a good performance when solving some large linear algebra problems, especially eigenproblems. This method is the most frequently one used for very large eigenproblems and it is also well adapted to massively-parallel architectures. The choice of the subspace size is very important both for numerical stability and speed-up. We conclude with a discussion on the effects of the orders chosen for the subspaces on performance.

A subspace forward iteration method for solving the quadratic eigenproblem associated with the FDE formulation

This paper presents the development of an efficient and numerically stable algorithm for accurately computing the eigensolutions of the quadratic real symmetric eigenproblem arising in the finite dynamic element (FDE) formulation. Closely related to the subspace forward iteration method, the proposed scheme is well suited to extracting the lowest natural frequencies and associated mode shapes of large practical eigenproblems, takes full advantage of the banded configuration of the stiffness, inertia and dynamic correction matrices involved in the eigenproblem and ensures a monotonic convergence from above to the required eigenpairs. A shifting technique for convergence acceleration and an eigenpair verification scheme are also presented. Numerical examples are shown demonstrating the excellent performance of the solution procedure.

An effective Lanczos algorithm for large eigenproblems.

An effective Lanczos algorithm for large eigenproblems.

On “a new splitting to solve a large hermitian eigenproblem”

On “a new splitting to solve a large hermitian eigenproblem”

Solution of large, dense symmetric generalized eigenvalue problems using secondary storage

This paper describes a new implementation of algorithms for solving large, dense symmetric eigen-problems AX = BX Λ, where the matrices A and B are too large to fit in the central memory of the computer. Here A is assumed to be symmetric, and B symmetric positive definite. A combination of block Cholesky and block Householder transformations are used to reduce the problem to a symmetric banded eigenproblem whose eigenvalues can be computed in central memory. Inverse iteration is applied to the banded matrix to compute selected eigenvectors, which are then transformed back to eigenvectors of the original problem. This method is especially suitable for the solution of large eigenproblems arising in quantum physics, using a vector supercomputer with fast secondary storage device such as the Cray X-MP with SSD. Some numerical results demonstrate the efficiency of the new implementation.

A partial preconditioned conjugate gradient method for large eigenproblems

Abstract In this paper a preconditioned iterative method suitable for the solution of the generalized eigenvalue problem Ax = λBx is presented. The proposed method is suitable for the determination of extreme eigenvalues and their corresponding eigenvectors of large sparse eigenproblems derived from the finite element discretization method. The solution is obtained through the minimization of the Rayleigh quotient by a preconditioned conjugate gradient (CG) method. The proposed triangular splitting preconditioner combines Evans' SSOR preconditioner with a drop-off tolerance criterion and is called partial preconditioner (PPR). The PPR is attractive in a large FE framework because it is simple and can be implemented at the element level, as opposed to incomplete Choleski preconditioners, which require a sparse assembly. Because of the renewed interest in CG techniques for FE work on microprocessors and parallel computers it is believed that this improved approach to the generalized eigenvalue problem, through the minimization of the Rayleigh quotient, is likely to be very promising.

Selection of Degrees of Freedom for Dynamic Analysis

A technique for the selection of dynamic degrees of freedom (DDOF) of large, complex structures for dynamic analysis is described and the formulation of Ritz basis vectors for static condensation and component mode synthesis is presented. Generally, the selection of DDOF is left to the judgment of engineers. For large, complex structures, however, a danger of poor or improper selection of DDOF exists. An improper selection may result in singularity of the eigenvalue problem, or in missing some of the lower frequencies. This technique can be used to select the DDOF to reduce the size of large eigenproblems and to select the DDOF to eliminate the singularities of the assembled eigenproblem of component mode synthesis. The execution of this technique is discussed in this paper. Examples are given for using this technique in conjunction with a general purpose finite element computer program GENSAM[1].

Solution of large eigenproblems on a microcomputer

A microcomputer program is presented for determining the lowest eigenfrequencies of large structural systems by using the finite element method. The solution procedure is based upon the subspace iteration technique. Householder's tridiagonalisation and the QL method are used in order to solve the eigenproblem in the reduced space at each iteration. The computer code has been implemented in BASIC and its present version is suitable for the solution of eigensystems with 1000–2000 degrees of freedom. The features of the program are pointed out and numerical tests are presented. It is shown that very accurate results can be obtained, although the computer system we have used sets a few constraints which often imply very long computing times (particularly when large bandwidth systems are given).

The application of gradient minimization methods and higher order discrete elements to shell buckling and vibration eigenproblems

Gradient minimization methods have been successfully applied to the discrete element analysis of shells with geometric nonlinearities and to plates with material nonlinearities. More recently, the feasibility of using such methods for the lower buckling and vibration modes of large eigenproblems has been established. The present investigation extends this work by developing scaling criteria and rescaling strategies based on the second variation of the discretized Rayleigh quotient. The importance of scaling to the efficiency of the conjugate gradient algorithm is found to be similar to that reported for potential energy minimization. Several shell vibration and shell buckling problems are than analyzed using the 48 degree of freedom Bogner cylindrical panel element reformulated to include an incremental stiffness matrix. This element is based on bicubic Hermite polynomials for which discretization error bounds indicate rapid eigenvalue and eigenvector convergence in certain elliptic boundary value problems. These convergence rates are numerically tested in both shell buckling and vibration eigenproblems.